# Conceptual explanation: Finding revolutions over a given time interval

The following question from NJCTL was assigned for me as homework. It is not a heavy computational problem, but rather, a conceptual one. I reviewed the meta guide regarding these types of questions and so I hope this is appropriate.

An object starts at rest and accelerates at a constant rate in a circular path. After a certain time $$t$$, the object reaches the angular velocity $$\omega$$. How many revolutions did it make over time $$t$$?

The supplied answer, in general terms of course, is $$\omega t/4\pi$$. I do not understand this type of problem conceptually and an internet search for this type of problem did no avail.

What made sense to me was to pick an arbitrary angular acceleration $$\alpha$$ and make a list of angular velocities.

My attempt with some casework:

Letting $$\alpha=3$$ revolutions$$/s^2$$ we have the following where $$t$$ is in seconds and $$\omega$$ is in revolutions per second.

($$t, \omega$$)

($$0,0$$)

($$1,3$$)

($$2,6$$)

($$3,9$$)

Over this interval of $$3$$ seconds, we have an angular displacement of 18 revolutions.

My problem arises here. Angular displacement seems to be growing at a different rate than the solution gives. I am not sure how $$\pi$$ makes its way into the solution if I begin with revolutions rather than radians. I am looking to intuitively understand how this solution is formulated. I believe that I took the wrong approach, however, I cannot think of any other way to move forward with this.

Thank you.

• Why are you beginning with revolutions? $\omega$ is usually given in radians/s. – BowlOfRed Mar 11 at 21:18
• I am aware of that. I was taught that $\omega$ given in revolutions is acceptable and so I started the problem with them since the question asks for number of revolutions. I did try the same method with radians, however, it leaves me with the same issue. @BowlOfRed – Gnumbertester Mar 11 at 21:31
• It seemed like you were asking specifically about the $\pi$ there. If you begin with radians but convert to revolutions, you're going to divide by $2 \pi$. – BowlOfRed Mar 11 at 21:50
• I see. I should probably make clear that I question $\pi$ because I began the problem with revolutions. @BowlOfRed Either way though, I get $36\pi$ for the amount of radians over 3 seconds. I do not see how the solution is derived from this (in general). – Gnumbertester Mar 11 at 23:07

## 1 Answer

Sometimes it can be easier to think of rotational problems as if they are linear problems.

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#rlin

You are looking for the distance travelled based on time and average velocity. The page above shows how this is written for both linear and rotational situations. The only difference between that and your question is the final conversion to revolutions instead of radians.